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Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)
 




Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

binomial theorem


 

binomial theorem


 

A. Binomial Theorem

The formula by which any positive integral power of a binomial expression can be expanded in the form of a series is known as Binomial Theorem. If x, y Î R and n Î N, then

(x + y)n = nC0xn + nC1xn _ 1y + nC2xn _ 2 y2 + ........+ nCrxn_ryr + ......+ nCnyn = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT).

This theorem can be proved by induction.

Observations :

(a) The number of terms in the expansion is (n + 1) i.e. one more than the index.

(b) The sum of the indices of x & y in each term is n.

(c) The binomial coefficients of the terms (nC0, nC1.........) equidistant from the beginning and the end are equal.

Ex.1 The value of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is

Sol. The numerator is of the form a3 + b3 + 3ab (a + b) = (a + b)3 where a = 18, and b = 7

Nr = (18 + 7)3 = (25)3. Denominator can be written as

Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = (3 + 2)6 = 56 = (25)3 Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

B. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE

(a) General term : The general term or the (r + 1)th term in the expansion of (x + y)n is given by

Tr+1 = nCrxn_r. yr

Ex.2 Find : (a) The coefficient of x7 in the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

(b) The coefficient of x_7 in the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Also, find the relation between a and b, so that these coefficients are equal.

Sol. (a) In the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT), the general terms is Tr + 1 = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

putting 22 _ 3r = 7 Þ 3r = 15 Þ r = 5 T6 = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Hence the coefficient of x7 in Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is 11C5 a6b_5.

(b) In the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT), general terms is Tr + 1 = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

putting 11 _ 3r = _ 7 Þ 3r = 18 Þ r = 6 Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)


 


Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)
 



 

Hence the coefficient of x_7 in Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is 11C6a5b_6

Also given coefficient of x7 in Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = coefficient of x_7 in Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ ab = 1 ( 11C5 = 11C6). Which is a required relation between a and b.

Ex.3 Find the number of rational terms in the expansion of (91/4 + 81/6)1000.

Sol. The general term in the expansion of (91/4 + 81/6)1000 is Tr + 1 = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

The above term will be rational if exponent of 3 and 2 are integers. i.e.Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) and Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)must be integers

The possible set of values of r is {0, 2, 4, ..........1000}. Hence, number of rational terms is 501

(b) Middle term : The middle term(s) in the expansion of (x + y)n is (are)

(i) If n is even, there is only one middle term which is given by T(n + 2)/2 = nCn/2. xn/2. yn/2

(ii) If n is odd, there are two middle terms which are Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) & Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Ex.4 Find the middle term in the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Sol. The number of terms in the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is 10 (even). So there are two middle terms.

i.e. Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)th and Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)th two middle terms. They are given by T5 and T6

T5 =T4 + 1 = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

and Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

(c) Term independent of x : Term independent of x contains no x ; Hence find the value of r for which the exponent of x is zero.

Ex.5 The term independent of x in Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is

Sol. General term in the expansion is Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

For constant term, Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Þ r=Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) which is not an integer. Therefore, there will be no constant term.


 


Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)
 



 

(d) Numerically greatest term : To find the greatest term in the expansion of (x + a)n.

We have (x + a)n = xn Binomial Theorem, Chapter Notes, Class 11, Maths (IIT); therefore, since xn multiplies every term in Binomial Theorem, Chapter Notes, Class 11, Maths (IIT), it will be sufficient to find the greatest term in this later expansion. Let the Tr and Tr + 1 be the rth and (r +1)th terms in the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) then Binomial Theorem, Chapter Notes, Class 11, Maths (IIT). Let numerically, Tr + 1 be the greatest term in the above expansion. Then Tr + 1 ³ Tr Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Substituting values of n and x, we get r £ m + f or r £ m where m is a positive integer and f is fraction such that 0 < f < 1. In the first case Tm + 1 is the greatest term, while in the second case Tm and Tm + 1 are the greatest terms and both are equal.

Ex.6 Find numerically the greatest term in the expansion of (3 _ 5x)11 when x = 1/5

Sol. Since (3 _ 5x)11 = 311 Binomial Theorem, Chapter Notes, Class 11, Maths (IIT). Now in the expansion of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT),

we have Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ 4r £ 12 Þ r £ 3 r = 2, 3

so, the greatest terms are T2 + 1 and T3+1. Greatest term (when r = 2)

Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

and greatest term (when r = 3) = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

From above we say that the value of both greatest terms are equal.

C. If Binomial Theorem, Chapter Notes, Class 11, Maths (IIT), where I & n are positive integers, n being odd and 0 < f < 1, then (I + f). f = Kn where A _ B2 = K > 0 & Binomial Theorem, Chapter Notes, Class 11, Maths (IIT). If n is an even integer, then (I + f) (1 _ f) = kn

Ex.7 If = [N] + F and F = N _ [N] ; where [*] denotes greatest integer, then NF is equal to

Sol. Since Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = [N] + F. Let us assume that f = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT); where 0 £ f < 1.

Now, [N] + F _ f = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) _ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Þ [N] + F _ f = even integer.

Now 0 < F < 1 and 0 < f < 1 so _1 < F _ f < 1 and F _ f is an integer so it can only be zero

Thus NF = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = 202n + 1.


 


Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)
 



 

D. Some Results on Binomial Coefficients

(a) C0 + C1 + C2 + ............+ Cn = 2n

(b) C0 + C2 + C4 + ............= C1 + C3 + C5 + .......... = 2n _ 1

(c) C02 + C12 + C22 + ............+ Cn2 = 2nCn = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

(d) C0. Cr + C1. Cr + 1 + C2. Cr + 2 +............+Cn _ r Cn =

Remember : (2n) ! = 2n. n! [1.3.5........(2n _ 1)]

Ex.8 If (1 + x)n = C0 + C1x + C2x2 +.................+ Cnxn then show that the sum of the products of the Ci¢s taken two at a time represents by : Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is equal to Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Sol. Since (C0 + C1 + C2 + ........+Cn _ 1 + Cn)2 Binomial Theorem, Chapter Notes, Class 11, Maths (IIT).......+

Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT). Hence Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Ex.9 If (1 + x)n = C0 + C1x + C2x2 + .........+Cnxn then prove that Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Sol. L.H.S Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = (C0 + C1)2 + (C0 + C2)2 + ....+ (C0 + Cn)2 + (C1 + C2)2 + (C1 + C3)2 + ....+ (C1 + Cn)2

+ (C2 + C3)2 + (C2 + C4)2 + ........+ (C2 + Cn)2 + .........+ (Cn_1 + Cn)2

= n (C02 + C12 + C22 +..........+Cn2) + Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) {from Ex. 8}

= n. 2nCn + 22n _ 2nCn = (n _ 1). 2nCn + 22n = R.H.S.

E. Binomial theorem for negative or fractional indices

If n Î Q, then (1 + x)n = 1 + nx + Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) + Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) +........... ¥ provided |x| < 1.

Note :

(i) When the index n is a positive integer the number of terms in the expansion of (1 + x)n is finite i.e. (n + 1) & the coefficient of successive terms are : nC0, nC1, nC2, .........., nCn

(ii) When the index is other than a positive integer such as negative integer or fraction, the number of terms in the expansion of (1 + x)n is infinite and the symbol nCr cannot be used to denote the coefficient of the general term.

(iii) Following expansion should be remembered (|x| < 1)

(a) (1 + x)_1 = 1 _ x + x2 _ x3 + x4 _ .........¥

(b) (1 _ x)_1 = 1 + x + x2 + x3 + x4 + .........¥

(c) (1 + x)_2 = 1 _2x + 3x2 _ 4x3 + .........¥

(d) (1 _ x)_2 = 1 + 2x + 3x2 + 4x3 + .........¥

(iv) The expansions in ascending powers of x are only valid if x is 'small'. If x is large i.e. |x| > 1 then we may find it convenient to expand in powers of 1/x, which then will be small.


 


Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)
 



 

F. Approximations

Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

If x < 1, the terms of the above expansion go on decreasing and if x be very small, a stage may be reached when we may neglect the terms containing higher powers of x in the expansion. Thus, if x be so small that its squares and higher powers may be neglected then (1 + x)n = 1 + nx, approximately,

This is an approximate value of (1 + x)n

Ex.10 If x is so small such that its square and higher powers may be neglected then find the approximate value of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Sol. Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = 1 _ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) _ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) = 1 _ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

Ex.11 The value of cube root of 1001 upto five decimal places is

Sol. Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

= 10 {1 + 0.0003333 _ 0.00000011 + ......} = 10.00333

Ex.12 The sum of Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is

Sol. Comparing with 1 + nx + Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)+ ...... Þ nx = 1/4 ...(i)

& Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) or Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) ...(ii) {by (i)}

putting the value of x in (i) Þ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) Þ n = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

sum of series = (1 + x)n = (1 _ 1/2)_1/2 = (1/2)_1/2 = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

G. Exponential series

(a) e is an irrational number lying between 2.7 & 2.8. Its value correct upto 10 places of decimal is 2.7182818284.

(b) Logarithms to the base 'e' are known as the Napierian system, so named after Napier, their inventor. They are also called Natural Logarithm.

(c) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) where x may be any real or complex number & Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)


 


Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)
 



 

(d) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) where a > 0

(e) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

 

H. Logarithmic series

 

(a) ln (1 + x) = x _ Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) where _ 1 < x £ 1

(b) ln (1 _ x) = Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) where _ 1 £ x < 1

Remember : (i) Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) (ii) eln x = x (iii) ln2 = 0.693 (iv) ln 10 = 2.303

The document Binomial Theorem, Chapter Notes, Class 11, Maths (IIT) is a part of Class 11 category.
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FAQs on Binomial Theorem, Chapter Notes, Class 11, Maths (IIT)

1. What is the Binomial Theorem?
Ans. The Binomial Theorem is a mathematical formula that provides a quick and easy way to expand or simplify a binomial expression raised to a positive integer power. It states that the nth power of a binomial expression can be expanded into a sum of n+1 terms, where each term is obtained by multiplying the binomial expression by itself n times, with each term being multiplied by a unique coefficient.
2. What is the importance of the Binomial Theorem in mathematics?
Ans. The Binomial Theorem is a fundamental concept in algebra and plays a crucial role in many areas of mathematics, including calculus, probability, and statistics. It is particularly useful in simplifying complex algebraic expressions, calculating probability distributions, and solving problems related to combinatorics. Moreover, it has practical applications in fields such as physics, engineering, and economics.
3. What are the applications of the Binomial Theorem in real-life scenarios?
Ans. The Binomial Theorem has numerous real-life applications, such as in calculating compound interest rates, determining the probability of success or failure in experiments, predicting outcomes in games of chance, and analyzing population growth rates. It is also used in cryptography to encrypt and decrypt messages, as well as in coding theory to construct error-correcting codes.
4. How can one use the Binomial Theorem to expand binomial expressions?
Ans. To expand a binomial expression using the Binomial Theorem, one needs to follow these steps: 1. Identify the values of n, a, and b in the expression (a+b)^n. 2. Write the Binomial Theorem formula, which is (a+b)^n = C(n,0)*a^n*b^0 + C(n,1)*a^(n-1)*b^1 + ... + C(n,k)*a^(n-k)*b^k + ... + C(n,n)*a^0*b^n 3. Substitute the values of n, a, and b into the formula. 4. Expand each term according to the binomial coefficient C(n,k), which is equal to n choose k. 5. Simplify the resulting expression by combining like terms.
5. What is the relationship between the Binomial Theorem and Pascal's Triangle?
Ans. Pascal's Triangle is a triangular array of numbers that is used to calculate the coefficients in the Binomial Theorem expansion of a binomial expression. The nth row of Pascal's Triangle corresponds to the coefficients of the nth power of the binomial expression (a+b)^n. Each number in the triangle is the sum of the two numbers directly above it, with the first and last numbers in each row being equal to 1. The values in Pascal's Triangle can be used to determine the coefficients in the Binomial Theorem formula, which simplifies the process of expanding binomial expressions.
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